Theorem txindis 22239

Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txindis	⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Proof of Theorem txindis

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4259	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
2		indistop 21607	. . . . . . . . . . 11 ⊢ {∅, 𝐴} ∈ Top
3		indistop 21607	. . . . . . . . . . 11 ⊢ {∅, 𝐵} ∈ Top
4		eltx 22173	. . . . . . . . . . 11 ⊢ (({∅, 𝐴} ∈ Top ∧ {∅, 𝐵} ∈ Top) → (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
5	2, 3, 4	mp2an 691	. . . . . . . . . 10 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
6		rsp 3170	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
7	5, 6	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
8		elssuni 4830	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ ∪ ({∅, 𝐴} ×t {∅, 𝐵}))
9		indisuni 21608	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
10		indisuni 21608	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐵) = ∪ {∅, 𝐵}
11	2, 3, 9, 10	txunii 22198	. . . . . . . . . . . . . 14 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ∪ ({∅, 𝐴} ×t {∅, 𝐵})
12	8, 11	sseqtrrdi 3966	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
13	12	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
14		ne0i 4250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑧 × 𝑤) → (𝑧 × 𝑤) ≠ ∅)
15	14	ad2antrl 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ≠ ∅)
16		xpnz 5983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅) ↔ (𝑧 × 𝑤) ≠ ∅)
17	15, 16	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅))
18	17	simpld 498	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ≠ ∅)
19	18	neneqd 2992	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑧 = ∅)
20		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, 𝐴})
21		indislem 21605	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
22	20, 21	eleqtrrdi 2901	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, ( I ‘𝐴)})
23		elpri 4547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {∅, ( I ‘𝐴)} → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
25	24	ord 861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑧 = ∅ → 𝑧 = ( I ‘𝐴)))
26	19, 25	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 = ( I ‘𝐴))
27	17	simprd 499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ≠ ∅)
28	27	neneqd 2992	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑤 = ∅)
29		simplr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, 𝐵})
30		indislem 21605	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐵)} = {∅, 𝐵}
31	29, 30	eleqtrrdi 2901	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, ( I ‘𝐵)})
32		elpri 4547	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {∅, ( I ‘𝐵)} → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
33	31, 32	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
34	33	ord 861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑤 = ∅ → 𝑤 = ( I ‘𝐵)))
35	28, 34	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 = ( I ‘𝐵))
36	26, 35	xpeq12d 5550	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) = (( I ‘𝐴) × ( I ‘𝐵)))
37		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ⊆ 𝑥)
38	36, 37	eqsstrrd 3954	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
39	38	adantll 713	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
40	13, 39	eqssd 3932	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵)))
41	40	ex 416	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
42	41	rexlimdvva 3253	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
43	7, 42	syld 47	. . . . . . . 8 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
44	43	exlimdv 1934	. . . . . . 7 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑦 𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
45	1, 44	syl5bi 245	. . . . . 6 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (¬ 𝑥 = ∅ → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
46	45	orrd 860	. . . . 5 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
47		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
48	47	elpr 4548	. . . . 5 ⊢ (𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ↔ (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
49	46, 48	sylibr 237	. . . 4 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))})
50	49	ssriv 3919	. . 3 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ {∅, (( I ‘𝐴) × ( I ‘𝐵))}
51	9	toptopon 21522	. . . . . . 7 ⊢ ({∅, 𝐴} ∈ Top ↔ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)))
52	2, 51	mpbi 233	. . . . . 6 ⊢ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
53	10	toptopon 21522	. . . . . . 7 ⊢ ({∅, 𝐵} ∈ Top ↔ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵)))
54	3, 53	mpbi 233	. . . . . 6 ⊢ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))
55		txtopon 22196	. . . . . 6 ⊢ (({∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) ∧ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))) → ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))))
56	52, 54, 55	mp2an 691	. . . . 5 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵)))
57		topgele 21535	. . . . 5 ⊢ (({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))) → ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵))))
58	56, 57	ax-mp 5	. . . 4 ⊢ ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵)))
59	58	simpli 487	. . 3 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵})
60	50, 59	eqssi 3931	. 2 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (( I ‘𝐴) × ( I ‘𝐵))}
61		txindislem 22238	. . 3 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
62	61	preq2i 4633	. 2 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} = {∅, ( I ‘(𝐴 × 𝐵))}
63		indislem 21605	. 2 ⊢ {∅, ( I ‘(𝐴 × 𝐵))} = {∅, (𝐴 × 𝐵)}
64	60, 62, 63	3eqtri 2825	1 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  {cpr 4527  ∪ cuni 4800   I cid 5424   × cxp 5517  ‘cfv 6324  (class class class)co 7135  Topctop 21498  TopOnctopon 21515   ×_t ctx 22165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-tx 22167

This theorem is referenced by: (None)